where \(\alpha (x_i)\)  captures the position-dependent strength of the nonlocal interactions. To obtain a well-posed and causal theory, it is required that \(\alpha (x_i)\in (0,1]\)  [1]}.
Notably, the continuum model with the power-law distribution of order \(1+\alpha (x_i)\in (1,2]\)  directly reflects the power-law dispersion relations [2]}, which is widely supported by experimental observations [3]}, [4]}, [1]}. Additionally, we also note that strength of long-range forces between spherical inclusions also exhibit power-law distributions with exponents in the interval \((1,2)\)  [6]}, [7]}, [8]}.
\(c_0\)  is a book keeping parameter defined in Eq. (REF ). The total internal force on the \(i^{th}\)  inclusion is given by:
\(F(x_i)=\sum _{j=-\infty }^{i-1}k_{ij}(u_i-u_j) + \sum ^{j=\infty }_{i+1}k_{ij}(u_j-u_i)\) 
